betsinda said:the line from the left approaches (1/2) at +1 and restarts at (1/2) at -1. There is a single point at (0,1/2)
No. It doesn't make any sense to talk about a point or an x value being differentiable. You can say, though, that a function is continuous or differentiable at a point or at some x value.betsinda said:would it then be correct to say that x=1/2 is not differentiable as it is not continuous at that point?
That is correct, so you are not misunderstanding the concept. I have made a couple of edits to what you wrote:betsinda said:based on the fact that well every function is not differentiable, very function that is differentiable is continous. Or am I misunderstanding the concept?
While not every function is differentiable, every function that is differentiable is continous.
The definition of continuity at a given point on a graph is that the function is continuous if the limit of the function exists at that point and is equal to the value of the function at that point.
You can tell if a graph is continuous at a given point by checking if there are any breaks, holes, or jumps in the graph at that point. If there are none, then the graph is considered to be continuous at that point.
For a graph to be differentiable at a given point, it means that the slope of the graph at that point exists and is defined by the derivative of the function at that point.
You can determine if a graph is differentiable at a given point by checking if the graph is continuous at that point and if the tangent line at that point exists and is not vertical. If both of these conditions are met, then the graph is differentiable at that point.
Yes, a graph can be continuous but not differentiable at a given point. This can happen if the graph has a sharp turn or corner at that point, which would make the tangent line undefined. However, if a graph is differentiable at a given point, it must also be continuous at that point.